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THE EINSTEIN THEORY 













































































THE 

EINSTEIN 

THEORY 

RELATIVITY AND GRAVITATION WITH 
SOME OF THE MORE SIGNIFICANT 
IMPLICATIONS 

For the General Reader 


By 

Gruber, D. D., LL. D. 

it 

Associate Editor, Bibliotheca Sacra. 


L. Franklin 


Author of “Creation Ex Nihilo: The Physical Universe a 
Finite and Temporal Entity,” “The Theory of a 
Finite and Developing Deity Examined,” 

“What After Death?” Etc. 


THE LUTHERAN LITERARY BOARD 
Publishers 

BURLINGTON, IOWA 
1923 


QCc= 

.Q-i 


Copyright, 1923 
by R. Neumann 
Burlington, Iowa 


NOV -6 '23 

©C1A760767 


. / l 


CONTENTS 


Preface 


PAGE 

.. 7 


PART ONE. 

The Special Theory of Relativity. 

I. Amm Motion Relative.15 

II. The Velocity of Light Unaffected by the 

Velocity of Its Source.23 

III. Time and Space Relative.26 

IV. Measuring-Rods Shortened and Clocks 

“Lengthened” (Going Slower) with In¬ 
crease in Velocity.32 

V. A Startling Deduction as to Energy.39 

VI. What Is Four-Dimensional Space?.41 

PART TWO 

The General Theory of Relativity. 

I. Provisional Statement of Theory.51 

II. Propositions of Euclidean Geometry Not 

Exactly Valid in Gravitational Field. .. .54 

III. The Necessarily Non-Euclidean Space-Time 

Continuum and the Use of Gaussian Co- 
Ordinates .57 

IV. An Entirely New Conception of Gravita¬ 

tion .60 

V. Light Deflected in Passing Through Gravi¬ 

tational Field.66 

VI. The Theory of Relativity Alone Explains 

the Anomalous Motion of the Orbit of 
Mercury.71 

VII. The Physical Universe a Finite Entity.... 74 

Index .83 


5 
















PREFACE 



HE revolution introduced into the 
physical conceptions of the world is 
to be compared in extent and depth 
only with that brought about by the 
introduction of the Copernican sys¬ 
tem of the universe.” This significant statement as 
to the Einstein Theory of Relativity was made by 
no less eminent a scientist than Prof. Dr. Max 
Planck of Berlin, a winner of the Nobel prize in 
physics. And Sir J. J. Thomson, president of the 
British Royal Society, gravely spoke of a test of 
the theory in its application to light passing 
through a gravitational field, as “the most im¬ 
portant result obtained in connection with the 
theory of gravitation since Newton's day." 1 In¬ 
deed, judging from the apparent logical consist¬ 
ency of the theory and certain startling deductions 
and inferences, as well as from the several remark¬ 
able verifications of it that have already been made, 


1 Address before the joint meeting of the Royal Society and the 
Royal Astronomical Society of Great Britain, held November 9, 1919. 
This meeting was called to hear the report of the findings of two 
eclipse expeditions sent out to determine whether rays of light from a 
star are deflected in passing through the gravitational field of the sun, 
and, if so, whether the amount of deflection corresponds to that pre¬ 
dicted by Dr. Einstein. 


7 





the statements of these two great masters in the 
department of physical science are perhaps no ex¬ 
aggeration. 

Such is the estimate of the theory under consid¬ 
eration, and that estimate may be regarded as 
fast becoming the verdict of leading men of sci¬ 
ence everywhere. It sets forth conceptions of the 
universe that probably mark the beginning of a 
new era of research and scientific progress. Not 
to speak of the competing essays for the Higgins 
Prize Contest, conducted by the Scientific Amer¬ 
ican, many articles, most of them on special points 
of the theory, as well as a number of books on 
the subject in general, have accordingly appeared. 
And yet, as a second successful scientific test has 
recently been made (the recent eclipse expedition) 
another discussion of the theory, based chiefly upon 
Dr. Einstein's own publications, setting forth its 
elements largely in the order in which he devel¬ 
oped them and giving some of its more important 
implications, may nevertheless be welcomed. 

That interest in the subject is general even 
in America, has been reflected also by the many 
references to it in the newspapers of the country, 
from the great metropolitan dailies to the small 
country town weeklies. And, indeed, what the 
average reader knows about it, has, generally 
speaking, come to him through news items and 
editorial comments. And these have commonly 
been either so fragmentary, or so misleading in 
giving largely only curious unverified inferences, 


8 


that the question in the popular mind, with but 
very rare exceptions, still remains, “What is it 
all about?” It is our purpose, therefore, to set 
forth this remarkable theory in sufficiently in¬ 
telligible terms to enable the general reader to 
understand at least what it is about and what 
some of the most significant logical deductions and 
applications are. 

We are familiar with the arguments of some 
of the writers who do not see their way clear to 
accept the theory as a whole, as well as with those 
of some who entirely reject it. And, indeed, we are 
not entirely convinced as to all of its elements. But 
that the theory is a logical deduction from prem¬ 
ises furnished by men like Michelson and Morley, 
Lorentz and Fitzgerald, Eoetvos, Riemann, Min¬ 
kowski, Gauss, Ricci, Levi-Civita, et al , even 
its strongest opponents cannot deny. And that 
the theory is thus partly built upon the re¬ 
markable investigations and scientific contribu¬ 
tions of such great masters, must not be made 
an excuse for refusing to give credit to Dr. Ein¬ 
stein for what is undoubtedly the most wonderful 
scientifico-philosophic structure of our time. It 
is for those who do not agree with Dr. Einstein 
first to disprove his 'premises; and then it is in 
order to reject his theory which is so consistently 
built upon them. But then, of course, the task 
devolves upon them also to explain the otherwise 
inexplicable physical phenomena which that the¬ 
ory alone accounts for. Meanwhile it is only 


9 


fair to Dr. Einstein, and indeed even to those who 
furnished him with some of the scientific and 
mathematical tools, that the public be furnished 
with as impartial a statement as possible of the 
actual elements of the theory upon which to base 
their candid judgment. And this, nothing more 
nor less, the writer proposes to do in the follow¬ 
ing pages. 

It should hardly be necessary to add that, in 
the setting forth of so important a theory for the 
general reader, in which as to certain special 
points at least it is quite natural to approach the 
form almost of a summary, we are of course in¬ 
debted to the one who is its distinguished author. 
We therefore take this occasion to acknowledge our 
indebtedness to Dr. Einstein in his own published 
statement of his theory, and to some extent also 
to certain of its best qualified expounders. 

It must be acknowledged, however, that it is 
not an easy matter to explain, to the entire satis¬ 
faction of perhaps many a reader, some of the 
more difficult points of so recondite and abstract 
a subject, especially within the scope of this dis¬ 
cussion. And yet, we believe that such points are 
developed with sufficient fulness and clearness to 
show at least their importance for the theory as 
a whole. However, for a fuller development of 
certain mathematical elements the reader is re¬ 
ferred to the more mathematical works on the 
subject, especially those of Dr. Einstein himself. 
But, after all, for the general reader the chief 


10 


interest in such elements is rather in the in¬ 
evitable results or conclusions reached from the 
premises than in the full processes by which these 
were attained. 

We believe that most of Part One should be 
quite easy to follow, as also should be at least the 
second half of Part Two , with its interesting appli¬ 
cations and far-reaching implications. But when¬ 
ever some point may at first seem difficult to grasp, 
it might be well for the reader to pass it over for 
the moment until further reading may throw light 
upon it. But no point that seems difficult should 
deter the reader from reading at once what is im¬ 
mediately intelligible, until the import of the the¬ 
ory as a whole may be grasped. 

It might also be said that certain of our in¬ 
ferences and applications may seem theoretical 
and curious rather than useful. But, even though 
we may not in all cases wholly accept them, we 
have set them forth in the interest of clearness, 
and to indicate that such are legitimate deduc¬ 
tions from the premises afforded by the theory. 
Moreover, in the discussion of different important 
points some repetition is almost unavoidable. But 
wherever such repetition occurs it is for the im¬ 
mediate fuller development of those points with 
as little reference to other points as possible. Our 
aim is clearness throughout, even though it might 
be at the expense of greater literary merit. 

In passing, it might even be stated that there 
is a certain apologetic value to the theory of rel¬ 


it 


ativity as developed by Dr. Einstein. It has ele¬ 
ments in it that unmistakably imply not only 
personality and free will in man, but also a neces¬ 
sarily existent Personality back of the universe 
and its phenomena. Jit issues in a finite universe, 
relative and dependent in every part, from the 
infinitesimal to the cosmic whole. And thus it ne¬ 
cessarily implies an independent, absolute and in¬ 
finite Entity, wholly different in essence, upon 
whose originating and sustaining power that de¬ 
pendent finite universe depends. / And that Entity 
must be a spiritual Personality, and of course more 
so rather than less so than the personality of man. 
We are therefore not surprised that Dr. Einstein 
in an ardent Zionist, a believer at least in the 
Jehovah-Elohim of the old Testament Scriptures. 
Such belief is in harmony with his theory. 

Although the General Theory of relativity is 
the more far-reaching and revolutionary, yet for 
a proper understanding of it a careful considera¬ 
tion of the Special Theory which led up to it, is 
necessary. This accounts for the following some¬ 
what fuller development of the latter than its rela¬ 
tive importance would otherwise demand. 

L. F. G. 

St. Paul, Minnesota. 

April, 1923. 


12 


Part One. 


THE SPECIAL THEORY OF RELATIVITY. 






































































5 


































• ' 































* 


































































































I. 

All Motion Relative. 


HERE was a time when everybody be¬ 
lieved the earth to be “standing still,” 
with its wondrous lights, the sun, 
moon and stars, moving from east to 
west across the face of the sky. Such 
a view involved, indeed, great difficulties even for 
the more thoughtful observers of that unscientific 
age. Not to speak of other difficulties, it was espe¬ 
cially hard, and indeed impossible, to account for 
the daily appearance of these heavenly bodies at the 
eastern end of the apparently flat face of the 
world. Did they travel over great passageways 
through or beneath the earth? And, if beneath, 
what held it in its apparently fixed place? Were 
they conveyed thither on the chariots of the gods ? 
Or did they for their journey eastward become in¬ 
visible to human eyes ? 

We almost smile at the very mention of such 
crude conceptions of the universe. But the human 
race did not arrive at a single bound at the rela¬ 
tively exalted views of the beginning of the twen- 



15 





tieth century. The progress of human thought 
from that age to this was a very gradual one. It 
was the result of perpetual groping through 
relative darkness toward the light. And as one 
world-theory was finally discarded and was suc¬ 
ceeded by another, it was always considered some¬ 
what strange that such discarded theory could 
ever have been held. And yet, when a new and 
better theory was first advanced, it was generally 
considered too strange, if not even absurd, to be 
given so much as a fair test. Who does not know 
of the difficulty with which the Copernican theory 
had to win its way! It was all too impossible 
to be tenable. The earth round like a ball, ro¬ 
tating on an axis and journeying around the 
sun ? People on the other side of the earth stand¬ 
ing with their heads downward and their feet 
upward ? Then why not fall off into space ? Then, 
too, what would sustain the earth upon its bot¬ 
tomless race-course? 

Such were the questions that had to be met 
and satisfactorily answered. Even a century 
after the Copernican theory was first given to 
the world a Galileo was compelled by a tribunal 
to renounce such ideas as ecclesiastically unortho¬ 
dox and in conflict with plain truth. But that 
the world is round and that it “does move,” is 
now accepted, yes, unconsciously taken for grant¬ 
ed as though it were an intellectual inheritance 
by every school boy and school girl. Nor are 
its once supposed difficulties any longer at all 


16 


perplexing. Indeed, it is not only found to be 
the only tenable explanation of the facts or phe¬ 
nomena, but it can in various ways be demon¬ 
strated to the satisfaction of even the untutored 
mind. 

How other now universally accepted theories 
won their way, space would forbid us to men¬ 
tion. The history of the progress of human 
thought is a long story, and in many respects it 
is stranger, more fascinating, than fiction. But 
we have given the above as an illustration that 
might be of service for our time. A new world¬ 
view is before the people of this generation fci 
consideration. It is indeed strange, and at first 
thought appears absurd in some of its elements; 
but surely not more so than was the theory of 
Copernicus, which has now been elevated to the 
plane of scientific fact. This age may, therefore, 
well profit by the history of Copernican astron¬ 
omy, and at least patiently wait for more light on 
such a scientific hypothesis as the Einstein theory. 

So smoothly or uniformly does the earth rotate 
on its axis and move through its orbit around the 
sun that it was impossible directly to discover 
that it is in motion at all. It was only by long 
and continued careful observations of the relative 
movements of the sun and other heavenly bodies 
through the sky, that this fact became apparent. 
At least the rotation of the earth might have been 
deduced from certain now well understood phe¬ 
nomena, but these phenomena themselves were 


17 


not carefully observed. And thus it was wholly 
immaterial for all practical purposes to the in¬ 
vestigators of that day whether the earth moves 
or “stands still.” Physical laws would assume 
the same form in either case. 1 * 

What was true for those earlier observers, 
and is even yet true if we ignore certain phenom¬ 
ena due to the motion of a point on the rotating 
and revolving earth deviating from a straight 
line, is true of uniform motion in a straight line 
everywhere. Such motion of a body cannot be 
detected by experiments upon that body itself 
alone. A man, waking up in a uniformly moving 
train in a railroad station, may attribute his ap¬ 
parent motion to the train on the next track, to 
his own train, or to both. Indeed, if no other ob¬ 
ject were visible, and if all windows and doors 
were closed, a man on such a uniformly moving 
train would not be conscious of any motion what¬ 
ever. It is thus only by observing other objects 
that such motion can be detected. And as for 
objects in such a moving train, like a falling apple 
or a weight hanging from the ceiling, their be¬ 
havior would be exactly the same as if the train 
stood still, unless disturbed by air currents 
through an open door or window. Thus we might 
say that whether the train were moving or stand¬ 
ing still, physical phenomena would be identical 

10 Of course, due to the deviation of the motion of an object on 
the rotating earth from a straight line, there are certain peculiar phe¬ 
nomena, such as that of the rotating pendulum in Foucault’s famous 
experiment in the Pantheon at Paris. 


18 



and therefore physical laws would assume the 
same form. Or, in other words, speaking phe¬ 
nomenally, it would make no difference whether 
the train were regarded as moving over fixed 
tracks or as standing still upon tracks (and 
earth) sliding beneath it. This is a concrete or 
special case illustrating what Dr. Einstein calls 
the restricted principle of relativity, which will 
be further developed and given a general form in 
what follows. 

Thus already from the preceding paragraphs 
it appears that all motion must be relative. It 
may be said even that if there were no other 
bodies in existence, or if there were none visible, 
we could never be conscious of any uniform mo¬ 
tion in a straight line even if it were with the ve¬ 
locity of light. Indeed, as we would apparently 
not be moving toward, away from or past any 
other body, all such motion would be meaningless. 
Again, if there were only one other body in ex¬ 
istence, no matter what its motion or what an 
observer’s motion, the other body would appear 
either stationary or as if it were moving away 
from or toward the observer, and always as if in 
a straight line. 

This view of motion is wholly at variance with 
the popular conception, according to which mo¬ 
tion has, so to speak, an absolute or independent 
existence. Indeed, there has been considerable 
confusion on this point in traditional mechanics. 
No really intelligible and accurate definition of 
19 


I 


motion can be given in terms of space coupled 
with time, as space itself is not really a some¬ 
thing definite or fixed, as we shall see. But mo¬ 
tion regarded as relative to some body of reference, 
or to a system of co-ordinates 2 rigidly attached 
to a body either relatively at rest or also in mo¬ 
tion, becomes intelligible and acquires a definite 
meaning. This is illustrated by a body falling 
from an aeroplane moving uniformly in a straight 
line. While the body would traverse a straight 
line in airless space relative to a system of co¬ 
ordinates rigidly attached to the aeroplane, it is 
found to describe a parabola relative to one at¬ 
tached to the ground. 

We said above in the illustration of the uni¬ 
formly moving train that physical phenomena 
would be identical and that physical laws would 
assume the same form whether the train were 
moving or standing still. We shall now give this 
principle a wider or more general application. If 
any two bodies, or systems of co-ordinates S and 
S' are both moving uniformly in straight lines as 
viewed from a relatively fixed point, then either 
one of these two bodies or systems of co-ordinates, 
as viewed from the other, would likewise be mov- 


2 If perpendiculars are dropped from a point to three plane sur¬ 
faces perpendicular to one another, and attached to a body, their 
measured lengths determine the position of that point with reference 
to those plane surfaces. These three perpendiculars are called the co¬ 
ordinates of that point, and are generally spoken of as Cartesian co¬ 
ordinates. This involves a rigid body to which such point has to be 
referred, while it takes for granted that the laws of Euclidean geom¬ 
etry hold for these measurements. 


20 



ing uniformly in a straight line, * * 3 although of 
course with a different velocity and in a different 
direction. Thus if the first system of co-ordin¬ 
ates 5 is a Galileian 4 system, the second system S', 
or any other system in uniform motion in a 
straight line with respect to S, must also be a 
Galileian system. Hence mechanical laws should 
hold good with respect to S' the same as for S, 
that is, natural phenomena should run their course 
according to the same laws with respect to both 
systems, as illustrated in the above - mentioned 
case of the behavior of objects in a train moving 
uniformly in a straight line. 

We spoke above of two motions of the earth, 
the diurnal motion on its axis and the annual mo¬ 
tion around the sun. And if it had no other 
motions, that is, if the sun were fixed or station¬ 
ary in space, there would be some meaning to the 
expression, “absolute motion of the earth.” For 
these two motions could accurately be determined 
from such stationary sun at any time. And even 
motions on the earth might then in turn be simi¬ 
larly treated. But, instead of being fixed, the 
sun itself is on a flight through so-called space, 
and the earth with ourselves upon its bosom is 
flying with the sun even while circling around it 
in its year-determining course. We look out in- 


8 Such motion with constant velocity and direction and without 

acceleration or rotation relative to a fixed body of reference, is spoken 

of as uniform translatory motion. 

4 A “Galileian” system of co-ordinates is one whose state of motion 
is such that the law of inertia really holds relative to it. 


21 



to space and find that during the course of years 
stars seem to be separating slightly in one direc¬ 
tion in the heavens and drawing closer together in 
the opposite direction. Hence we must apparent¬ 
ly be moving toward the one and away from the 
other. 

But investigation shows that this is so only in 
a general sense and that the stars move relatively 
among themselves. Therefore not even do the 
stars furnish us with a fixed point in space from 
which to measure direction and velocity. Then 
whither are we bound and what is our velocity of 
travel on the way? Do we journey in some par¬ 
ticular direction among the stars, or do the stars 
drift toward and by us in a gigantic sweep toward 
some celestial borderland? The truth, confirmed 
by an overwhelming accumulation of evidence, 
is, that both sun and stars—of which our sun is 
one—are “on the move,” and indeed that not only 
the stars but everything from electron to star is 
in ceaseless motion. Then what about a fixed 
point from which to determine so-called absolute 
motion? Even all cosmic motions are relative, 
and are either slow or fast in this direction or in 
that, according as they are measured from some 
other moving body or bodies. And this fact, that 
all motion throughout the universe is relative, is 
a fundamental principle underlying the Einstein 
theory. ) 


22 


II. 


The Velocity of Light Unaffected by the 
Velocity of Its Source. 



[N seeking for a sort of framework or 
body of reference with respect to 
which cosmic, and even other, mo¬ 
tions might be measured, the hypo¬ 
thetical ether was put to the test as 
a last possiblity. It had already been made to 
carry so many scientific burdens that it was 
pressed into service. If this medium exists, it 
must either drift with the earth, that is, the earth 
must float in its own garment of ether, or the ether 
must drift through the earth, and thus through all 
our instruments used in experimentation, even as 
water flows through a screen or through the inter¬ 
spaces of a pile of bullets, or as the air would flow 
through a moving open train. And thus, in the 
latter case this ether current should affect our in¬ 
struments and be measurable. 


But that the ether does not drift with the earth, 
as though the earth carried it with itself in its 
course, is generally accepted as experimentally es¬ 
tablished. And if the ether drifted through the 


23 




earth, then light approaching us from a point 
toward which we are moving should apparently 
have greater velocity than light approaching us 
from the opposite direction or even at right an¬ 
gles to our direction of motion. To determine 
whether this is the case, or really whether the 
ether has a motion relative to the earth, was the 
purpose of the remarkable light-experiment of 
Michelson and Morley in 1887, and again later by 
Morley and Miller with a larger interferometer, of 
which we can here not give details. But the re¬ 
sult of this experiment indicated that the velocity 
of light is apparently the same (186,300 miles per 
second) regardless of whether the observer ap¬ 
proaches or recedes from the light, or whatever the 
relative velocity of the source of light; but, as we 
shall see, this would strictly be true of light only 
in free or gravitationless space. Their instru¬ 
ments remained unaffected by any supposed ether 
drift. 

That the velocity of light should be constant 
regardless of the velocity of its source, or of that 
of the observer, is, however, so in conflict with the 
accepted principle of the addition of velocities in 
classical mechanics, that an explanation had to be 
sought. It even reopened the whole question as to 
the reality or existence of the ether. Finally Fitz¬ 
gerald, as also Lorentz, came forward with a 
plausible explanation, namely, that in compensa¬ 
tion for the relative velocity of light there must be 
a relative contraction or shortening of bodies in 


24 


the line of motion through the hypothetical ether, 
and that this could not experimentally be detected 
because even our measuring rods suffer the same 
ratio of shortening. Thus the same velocity of 
light would always be arrived at from experiment 
regardless of the relative motion of the source of 
light. 

That the velocity of light should be the 
same regardless of the relative velocity of its 
source, is in accord with Einstein's principle stated 
above, according to which the law of the trans¬ 
mission of light (in vacuo) should actually be the 
same for a reference body in uniform motion in a 
straight line as for one that is relatively fixed. And 
thus the remarkable earlier investigations of 
Michelson and Morley prove to be a striking con¬ 
firmation of Einstein's restricted principle of rela¬ 
tivity, as indeed they became a legitimate founda¬ 
tion for the whole theory. Moreover, this conclusion 
was reached without any regard to the existence 
of the ether, which does not at all enter into Ein¬ 
stein's scientific structure because it is not needed 
to account for any of its factors. It takes no ac¬ 
count of any such hypothetical medium either for 
transmission of light, etc., or for a something fixed 
from which to measure motion of ponderable mat¬ 
ter. All motion is regarded as only relative to 
bodies relatively in motion. 


25 


III. 

Time and Space Relative. 



Y A very simple illustration of a train 
moving between and in line with two 
supposedly equally distant bolts of 
lightning or light-signals, Dr. Ein¬ 
stein shows that events simultaneous 
with reference to the ground are not absolutely so 
with reference to a moving train (and vice versa). 
Or, in general, two events simultaneous for one 
person may not be so for another. What to one 
person is the relatively earlier of two events might 
be the relatively later to another person. Thus 
every reference-body, or co-ordinate system, is 
seen to have its own particular time, or the state¬ 
ment of time has no absolute meaning. 


The statement of time is seen to be dependent 
upon the state of motion of the body of reference. 
And because the time for a particular event with 
respect to the train cannot be the same as that of 
the same event as observed from the embankment , 
it follows that a person walking in a train does 
not traverse a certain distance relative to the 


26 






ground in a time equal to the time as measured 
from the ground, or he does not traverse the same 
distance relative to the embankment in a time 
equal both to an observer on the train and to one 
on the embankment. And thus from the fact of 
the relativity of simultaneity the relativity of the 
conception, of distance is readily deduced. Hence, 
it would follow that the length of the train as 
measured from the embankment should be different 
from its length as measured in the train itself. 
And so the distance covered in a unit of time by 
a man in a train as measured on the train, would 
also apparently not be equal to the distance as 
measured from the embankment. However, as will 
appear later, this could not experimentally be de¬ 
tected because all our instruments of measure must 
be relatively equally affected. Indeed, the length 
of an object cannot be definitely stated except in 
connection with a body of reference and is thus 
dependent upon relative motion and varies with 
change of motion. 

The apparent difficulty in the application of the 
theorem of the addition of velocities in traditional 
mechanics, especially to the propogation of light, 
as illustrated above, as Dr. Einstein points out, is 
therefore due to the unwarranted assumption that 
the time-interval between two events, and the 
space-interval between two points, on a rigid body, 
are independent of the condition of motion of the 
body of reference. And without this assumption 


27 


the theorem of the addition of velocities in classical 
mechanics becomes invalid. 5 

From the above considerations it is thus seen 
that both time and space are relative , and that they 
are closely associated with relative motion. In¬ 
deed, as will become even still more manifest, 
neither time nor space seem to have an independent 
individual existence, but are only different aspects 
of a greater whole. Thus there could, strictly 
speaking, be no absolute measure of either space 
or time, nor could there be any simultaneity to 
events happening at different places. Not only 
must all space- or time-measures be only relative 
to other space- or time-measures, but space- and 
time-measures must be relative with reference to 
each other. Hence, time, space, distance, not to 
speak now of mass and energy, are not only not 
the same for different observers, but even not for 
the same observer differently located. Indeed, 
considered separately, space and time, as well as 
direction, are only local affairs, relative to the ob¬ 
server. The observation of any event in space ne¬ 
cessarily takes place at or involves a certain time; 
the observation of any moment of time is asso¬ 
ciated with space. Hence time and space cannot 
be separated in a full description. Therefore, for 
a complete description of any event the element of 

5 One would, of course, suppose, if a train traveled with a velocity 
of twenty miles' per hour and a man walked on the train at the rate 
of three miles per hour* in the direction of travel, that the velocity of 
the man relative to the ground would be twenty-three miles per hour. 
In classical mechanics this would be correct. But that it is not strictly 
correct, according to the discussion above, will become more apparent 
from what follows. 


28 



time becomes a necessary factor, even as every 
point on a trajectory would mark a different mo¬ 
ment. Thus a point defined by four co-ordinates 
—three for ordinary space and one for time— 
marks the space-time of an event at a particular 
moment; and a succession of points (or a line) 
in space-time, describes its life-history. Such a line 
is called its “world-line” by Minkowski. 5 * 

In further demonstration and development of 
the points made above, it becomes necessary from 
the known place and time of an event with refer¬ 
ence to one body of reference relatively at rest to 
determine the place and time of the event with 
reference to another reference-body relatively in 
motion , with such a relation between place and 
time of an event relative to both reference-bodies 
that rays of light would have the same velocity of 
transmission with reference to both. Let us as¬ 
sume a co-ordinate system S at rest and another S' 
relatively in uniform rectilinear motion with the 
velocity v with the axis x of £ parallel to x' of S' 
Then any event may be fixed with respect to S by 
the co-ordinates x y y, z y t, while the corresponding 
co-ordinates x' y y' f z' y t' y would fix the same event 
with respect to S', the velocity of light being repre¬ 
sented by V. Now assuming that we know the 
values of x, y , z y t, of the event (with respect to S), 
the values of x', y', z', t', of the same event (with 


r '* An event may be said to be the arrival of a point in a given 
position and thus to be marked by the intersection of several world¬ 
lines. 


29 




respect to S') , as determined by Lorentz and called 
the “Lorentz transformation,” are as follows: 

x-.vt 

y' y; 

Z' =Z; 


From the above it is found that the law of the 
transmission of light in vacuo is satisfied for both 
S and S'. 1 It is seen that a general law of nature 
will remain unchanged by a change in treatment 
from one co-ordinate system to another in uniform 
rectilinear motion or by a change in the space- 
time variables of one such system of co-ordinates 
for those of another; or, all systems of co-ordi¬ 
nates in uniform motion in a straight line give us 
exactly the same form for the expression of the 
general laws of nature. This is a crystallized 
statement of the special theory of relativity. 

6 If the absolute character of time and space were assumed, as in 
the older mechanics, or if the velocity of light V were regarded as 
infinite, the above equations, it will readily be seen, would become 
the so-called “Galilei transformation,” as follows: 


X' 

¥ 


X-vl 

= V 
- z 
= t 


7 Dr. Einstein arrives at this conclusion by a very simple process. 
Light passes along the positive x-axis, so that x —Vt. Or'light travels 
with the velocity V. This relation' between x and t involves a similar 


30 







relation between x' and t'. If now Vt be substituted for x in the first 
and fourth equations of the Lorentz transformation given above. 


x' = (V- V ).t 

~\l*~ v* and *V - v' 

And from these equations, x' —Vt'. Hence the velocity of trans¬ 
mission with reference to S' is V, the same as with reference to S 
and of course for any particular ray of light, and therefore for light 
advancing in any direction. 



31 






IV. 


Measuring-Rods Shortened and Clocks “Length¬ 
ened” (Going Slower) with Increase 
in Velocity. 



j|ROM the above considerations of 
the relativity of space and time, it 
was already inferred that the lengths 
of measurings-rods, and the time of 
clocks, in motion should naturally 
share in that relativity. Indeed, such an inference 
is only a logical application of the principle above 
developed. 

From the first Lorentz equation, by a simple 
calculation, it is found that, as a yard-stick moves 
with a reference-frame S' in the direction of its 
length with velocity v relative to S, its length as 
measured from 5 relatively at rest must be only 


v 


1 - 


V 2 


of a yard as measured on S'. The 


yard-stick is therefore shortened when in motion as 
measured by an observer relatively at rest; and 
if its velocity v could become that of light V, the 


length of the yard-stick 



would become 


32 










zero, as will be seen by substitution, or it might be 
said to have vanished. Hence the velocity of light 
must be regarded as the limiting velocity, which 
can never be exceeded by any body of matter, be¬ 
cause the above expression for the length of the 
yard-stick would become negative , or rather imag- 
nary, if its velocity v were greater than that of 
light V. And what is true of the yard-stick in mo¬ 
tion on S' relative to S at rest, is of course equally 
true of it on S at rest as viewed from S' in motion . 8 

From the above it is seen that for great veloci- 
ites relative to any observer the shape of any 
body should be changed for that observer, however 
infinitesimally, every element of its line of motion 
being shortened in that direction so as to be only 



2 


V V 2 of itself when viewed relatively at rest, 
while elements perpendicular to that line are unaf- 


8 It will be noticed that, although Einstein uses the same formulae 
as Dr. Lorentz to determine the so-called contraction or shortening of 
an object in the line of motion, his explanation is different from that 
of Lorentz. While Lorentz ascribes it to the object, Einstein; ascribes 
it to the observer as it must appear to him according to the laws of 
the propagation of light. According to Lorentz the shortening is real, 
according to Einstein it is only apparent. According to Lorentz it 
could take place only in an object really in motion (if there were 
any such thing) ; according to Einstein it would take place for the 
observer whether he were observing an object supposedly in real motion 
from one really at rest or whether he were observing an object sup¬ 
posedly at rest from one supposedly in motion. 

However, even the idea of shortening in the line of motion 
should not seem strange in the light of the electromagnetic theory of 
matter. If the individual elements (electrons) which constitute matter 
are apparently shortened when moving with great velocity, we should 
expect that matter thus constituted should also be subject to the same 
law. But this is not the explanation of Einstein, nor could it likely 
be the correct explanation as the contraction is the same for all sub¬ 
stances of whatever composition. Lorentz's hypothesis is therefore not 
a part of the Einstein Theory of Relativity, according to which the 
whole is only phenomenal due to relative motions and the law of 
light. 


33 




fected. For a velocity of nine-tenths that of light 
a body would be shortened in the direction of its 
motion by over one-half, whereas for the earth 
with its velocity of about I 8 V 2 miles a second the 
shortening at the middle would be only about two 
and one-half inches. If a body could move faster 
than light it would not only lose all its length, but 
it might be said to appear reversed so as to have 
what might be called negative length. But as a body 
can never attain the velocity of light because its 
energy would then, according to our formula for 
energy, become infinite, we need not further con¬ 
sider this point. However, it might be said to have 
escaped wholly into the fourth dimension, a point 
of which we shall speak later. Hence, according 
to the theory of relativity, no body can be regarded 
as absolutely rigid. For the same reason, a man 
walking at the rate of two miles an hour on a 
boat that goes three miles an hour up stream, 
would not walk five miles, but 4.9999999999999999 
miles, as measured from the shore, as has been 
calculated. 

By direct measurement these results could, 
however, not be verified, because our very meas¬ 
ures when applied would partake of the same rel¬ 
ative shortening, as already noted. This has been 
well illustrated with a reflecting-spoon, or with an 
individual’s reflection in a curved mirror, in which 
case the measure will always be affected in the 
mirror exactly as is the imaged person about being 
measured. 


34 


Such a relative shortening alone would seem 
to make intelligible the result of the Michelson and 
Morley experiment, namely, that the velocity of 
light remained the same whether measured in the 
direction of—away from, or toward—the source 
of light, or even across the path of the earth's 
orbit, of which we have already spoken. 

Moreover, from the first and fourth Lorentz 
equations, it is found that, as a clock as judged 
from S is moving with the velocity v, as judged 

from 5 the interval between two ticks is 

seconds, or more than a second. Or the clock 
would go slower than when* it is at rest. This can 
readily be verified by inserting the velocity for v 
and that for V (186,300 mi.) in the formula and 
working out the value. 

It is thus seen that the amount of duration 
between seconds of the clock on S' in relative mo¬ 
tion, as viewed from or measured by the clock on 
S relatively at rest, would depend upon the relative 
velocity of S'. And as that velocity would approach 
the velocity of light, according to the above for¬ 
mula, the duration or interval between two success¬ 
ive ticks would approach infinity. So also would 
there be a similar lengthening out of the interval 
between any other successive events, or a propor¬ 
tional slowing down of all activity. Hence at the 
velocity of light there would be no recurring ticks 
of a second, or time relative to £ would cease. Or, 
if a clock were moving away from an observer 


35 


with the speed of light, its pendulum, hands, and 
all associated events, would for him stand still. 
And the same would, of course, also be true of the 
clock or events on S relative to S'. 

Thus a person traveling away with the speed 
of light would apparently never grow older to an 
observer remaining behind. Now if a body could 
attain a velocity greater than that of light (which 
it cannot, as we have seen), time might be said to 
be reversed so that a man might appear to get 
younger as well as smaller, etc. From this also it 
would seem that the velocity of light must be the 
limiting velocity. It is the nearest approach to 
instantaneity possible in a physically constituted 
universe. And, indeed, from certain implications 
of the theory of relativity, it is difficult absolutely 
to deny it a sense of instantaneity. 

From a comparison of the above calculations 
as to the apparent shortening of a body in the di¬ 
rection of its motion and the apparent lengthen¬ 
ing out of time, with relative velocity, it is seen 
that the one exactly balances the other, the length¬ 
ening out of the latter or the slowing down of 
activity being mathematically equal to the short¬ 
ening of the former. They are reciprocals of each 
other. Thus it would appear that with great ve¬ 
locity a body would be passing, according to a fixed 
law, partly into a so-called fourth dimension, the 
amount being exactly equal to that by which it 
seems to be shortened or to be passing out of three- 


36 


dimensional space. Hence, from this it becomes 
still more evident that space and time are so re¬ 
lated that the one varies in some way inversely 
with the other, and that neither can be separated 
from the other, the two together constituting a 
sort of indivisible space-time continuum of four 
dimensions. 

It might therefore be said that there is a sort 
of apparent conservation in the magnitude of an 
object, or in the duration of an event, in its space- 
time , as already implied. Or, while time and. 
space , considered separately, are relative to the 
observer, the so-called interval between two 
events in space-time is absolute. Just as an in¬ 
definite number of right triangles may have legs 
different in length and yet have the same hypot¬ 
enuse, so space-time may be divided in different 
ways into its time and space components, and yet 
always continue to have the same so-called space- 
time interval between the events. Change of mo¬ 
tion changes the division of the space-time into 
time and space, even as moving one’s position 
causes different divisions of a surface into right 
and left. Indeed, as each individual would di¬ 
vide the space-time continuum differently , there 
would be a r^al sense in which every person would 
be the center of the universe as he sees it. 9 

9 If the reality as to the universe corresponded actually to its ap¬ 
pearance to our senses, or if it were strictly a three-dimensional 
Euclidean entity with time as simply its separate measure of duration, 
instead of a four-dimensional space-time continuum, as relativity de¬ 
mands, then there would, of course, be no apparent shortening of a 
body in the line of motion nor this lengthening out of time, as viewed 


37 



by an observer relatively at rest. Hence an object (or event) moving 
away from an observed, with the velocity of light, would always appear 
(if visible) as it was ati half its age. Or a person born on a receding 
ray of light when it left an observer would fifty years afterwards 
appear as he was at twenty-five years of age. That is, he would appear 
as he was at the time the last ray of light left him for the observer, 
and this would take the same time that it took him to get to that 
point in space. Therefore clocks and events would not appear to 
stand still, but might be said in a sense to go half as fast as those 
about the observer. This illustration will indicate a striking differ¬ 
ence between the Euclidean or Newtonian view of the world and the 
Einsteinian view. 


/ 


\ 


\ 


38 




A Startling Deduction as to Energy. 



S a necessary deduction from the 
theory as so far developed it is seen 
that for relativity rapid motion 
classical mechanics would also have 
to be modified as to the subject of en¬ 
ergy. Dr. Einstein shows that according to the 
theory of relativity, a body moving with velocity v, 
absorbing energy e in the form of radiation as 
judged from a system of co-ordinates moving with 
the body, has its kinetic energy increased by 


/i-v 2 . From this it is seen that the resultant en- 
V V 2 jnV% (m*Aa)V* 

ergy of this body really becomes /y : ^ or r===F * 

v V s V V' 

This is the same as that of a body with mass 

TIUe. 

} the inertial mass, which is not constant, 
increasing by ^ 2 . 


It w T ill also be seen from these formulae that as 
the velocity v of the mass approaches V, the 
velocity of light , its kinetic energy also approaches 


39 











infinity . 10 And at the velocity of light there is no 
distinction between mass and energy. Indeed, 
inert mass may then be said to be latent energy. 
Hence mass and energy must apparently be equal , 
or perhaps better identical. The so-called law of 
the conservation of mass thus really becomes iden¬ 
tical with that of the conservation of energy. And 
energy is not strictly a constant qantity. 10 * The law 
of the conservation of energy is apparently only 
approximately true, although deviations from this 
law in experience are generally too small to be 
observed. 

10 From the above also it is apparent that the velocity of light plays 
the part of limiting velocity in nature, so-called gravitation itself act¬ 
ing with no greater velocity than that, as becomes more evident from 
the General Theory. Thus there can be no instantaneous action, either 
at a distance or through an hypothetical medium, unless the velocity 
of light be considered as infinite or instantaneous, which would be in 
conflict with our idea of infinity. 

10 * See also “Whence Came the Universe?” Pp. 173-178; 205 sqq. 


1 


40 



VI. 

What Is Four-Dimensional Space ? 



T has already been seen that time 
is inseparable from space, as is es¬ 
pecially apparent from the Lorentz 


equation 






and that it 


enters the space-time equation as a fourth dimen¬ 
sion of Minkowski’s world, which may be regarded 
in a formal manner as a four-dimensional Eu¬ 
clidean space with an imaginary time co-ordinate. 

It is difficult, and indeed impossible, for us to 
conceive of four-dimensional space. Because our 
sense-experience in space has not afforded us a 
four-dimensional idea, we cannot abstractly form. 
one. Our idea of space is built upon our experience 
with, or observation of, conditioning matter, with¬ 
out which space would have no form. But the fact 
that our senses do not give us an idea of such 
higher space, would not be conclusive evidence 
against such a reality; for they are developed upon 
and attuned to our immediate physical environ¬ 
ment, and can surely not be exhaustive of reality. 


41 








Such realities as do not come within the range 
of sense-experience and furnish a basis for sense 
development, can surely not be appreciated or ap¬ 
prehended by these sense-organs. And among such 
are properties of such higher space. But the very 
fact that they can be mathematically indicated, 
would seem to point to their existence. The fact 
is that what is regarded as the fourth dimension 
(time) is in effect the same for all persons, the 
time-scale or standard being in experience invari¬ 
able, whereas the three dimensions of Euclidean 
space as determined by so-called matter, or by 
objects in it, vary under different circumstances 
or with different objects for different individuals, 
and also for the same individual. Hence, while 
the three dimensions of Euclidean space have 
entered our consciousness as a reality, time 
as a fourth variable dimension has not, and 
can therefore not be visualized. But, according 
to the theory of relativity, it does vary with a 
body's velocities, as we have seen. That time 
might be regarded as a fourth dimension was sug¬ 
gested already by d’Alembert in 1754, and worked 
out more definitely by Minkowski in 1903. We 
may perhaps form some idea of four-dimensional 
space from illustrations. 

As we have seen, for a complete description 
of an event a fourth co-ordinate is necessary, the 
element of time entering into our equation. Hence 
the co-ordinate system of three-dimensional space 
might be regarded as uniformly moving along a 


42 


fourth co-ordinate time t. Similarly, with a point 
in motion. The three familiar, more usually so- 
called, dimensions —applied to so-called solids— 
in which we move about or have "elbow room,” 
constitute ordinary space; the duration or dura¬ 
tional direction in which such movements take 
place, or the invariable onward sweep of duration 
of that three-dimensional space, constitutes time. 
Thus time may be regarded as the constant uni¬ 
form flow of objects, with their associated events, 
in four-dimensional space, each cross-section be¬ 
ing the present, with past and future respective¬ 
ly behind and before. It is, indeed, this dura¬ 
tional flow of objects that is the basis of events 
or that makes them the possible, and indeed in 
another sense the necessary, associates of objects. 

Moreover, with equations of four co-ordinates 
or of four-dimensional space, a super-mind could 
express the whole universe to the last detail, ex¬ 
cept insofar as life and mind would invalidate 
those equations. Strange to say, only too often 
this last “exception” is disregarded, and often 
by men who should know better. Indeed, free 
will partly determines its space-time as through 
time it acts in three-dimensional space. To a per¬ 
sonality there is no fixed world of time and space, 
but a relative determination of the same with 
time. Free will and a relatively chosen future 
go hand in hand. 

A few very suggestive concrete illustrations 
of four-dimensional space may be cited. Imagine 


43 


a great number of flat (two-dimensional) sec¬ 
tions of a movie film, representing a series of 
scenes stretching over many years, laid on top 
of one another. Then the thickness of the pile 
of film sections would be a third dimension to the 
film as a whole. But as each scene would appear 
upon the screen and be interpreted as suposedly 
a three-dimensional reality, the thickness of the 
pile, or a cord holding them together chronologic¬ 
ally, would represent what might be called a 
fourth dimension, namely, time. Now if we 
could see all the sections (scenes) at once, the 
time represented by their series of years would 
be a moment. What we should ordinarily see in 
succession, we should now see simultaneously. So 
might a four-dimensional being see all past and 
future as continually present. The action of the 
machine might even be reversed and the film pre¬ 
sented backwards. 

A similar illustration might be made of a 
series of pictures taken one a day throughout the 
life of an individual, if conected by a line. Each 
picture would represent a scene in three-dimen¬ 
sional space, and the connection of these would 
represent the time co-ordinate. 

Again, imagine a circular cone (with axis 
perpendicular) rising out of the water with uni¬ 
form velocity. It would produce a uniformly in¬ 
creasing circle at the water’s level. A being lim¬ 
ited to the surface, would see only the ever-widen¬ 
ing circle , whereas three-dimensional beings like 


44 


ourselves would also see the rising cone. So our 
limitations with reference to a fourth dimension 
might be analogous to those of a two-dimensional 
being with reference to a third dimension, as il¬ 
lustrated in the rising cone. Moreover, if the 
cone were inclined and if the water were moving 
up and down instead of the cone, the elliptical 
cross-section would move along the water's surface 
but the two-dimensional creatures would ascribe 
this motion to the cone's cross-sections themselves 
instead of to the moving water-surface, and would 
be much puzzled by the change in size. 

A two-dimensional curved surface is curved in 
the third dimension, but a two-dimensional being 
could not conceive such curvature into a third 
dimension; nor could a three-dimensional being 
conceive of a curvature into a fourth dimension of 
space. As already noted, there was a time when 
the whole race lived under a similar delusion with 
reference to the earth, namely, that it must be 
flat. And yet the actual measurement of an ex¬ 
tended circle on a plane surface of the earth might 
have proved to investigators of that earlier unsci¬ 
entific age that the ratio of the circumference to 
the diameter would be less than 3.1416. They 
might thus almost have arrived at the truth, and 
even perhaps have calculated the amount of curv¬ 
ature into the third dimension, and therefore the 
magnitude of the earth. But so imperceptible is 
that curvature that it escaped their observation. 
In like manner may we now be laboring under the 
delusion that space is three-dimensional, while 


45 


its so-called three dimensions may curve into, 
and be part of, four-dimensional space. 

It has even also been contended that a four¬ 
dimensional being could not be excluded from a 
three-dimensional space enclosure like a room, 
even as a two-dimensional enclosure could not 
exclude a three-dimensional being (like a bird). 
Moreover, points far apart to a three-dimensional 
being might be very close to a four-dimensional 
being, even as are points at opposite ends of a bent 
surface, which would be far apart to a two-men- 
sional being, or as a great distance might sepa¬ 
rate two synchronous events. 

Such ideas seem strange to us, but if we were 
more accustomed to them they would probably 
gradually lose their strangeness. When the world 
was believed to be flat, a round world was as incon¬ 
ceivable as now are such ideas. Then up and down 
were supposed to be absolute directions, the same 
throughout the universe. But with the knowl¬ 
edge of a round world, with people all around 
it in every direction, not only the terms up and 
down, but even all other directions, have ac¬ 
quired a relative meaning. So with the newer 
world-view, even space itself becomes relative, 
while matter and space are found even to deter¬ 
mine each other, as will be seen later. We have 
simply risen to a broader generalization in our 
search for the reality lying behind the phenom¬ 
enal universe, wherein all things are relative; 


46 


\ 


and the consequent new strange conceptions will 
more and more become common and familiar. 

The four-dimensional conception of space is 
especially important for the understanding of, 
as it in a sense is fundamental to, the General 
Theory of Relativity , which we shall now pro¬ 
ceed to set forth. 


N 


47 











Part Two. 


THE GENERAL THEORY OF RELATIVITY. 


































































































































Provisional Statement of Theory . 



HUS far we have been considering 
what is properly called the Special 
Theory of relativity, because it per¬ 
tains only to the special case of ref¬ 
erence-bodies in uniform rectilinear 
motion. We have seen that either of two bodies, one 
relatively at rest and the other in uniform motion 
in a straight line, may be chosen as reference-body 
for an event. And as either reference-body may be 
considered in motion only with reference to the 
other, the general laws of nature would have to 
be the same for both. This reasoning assumes 
the existence of a moving reference-body with re¬ 
spect to which the law of inertia really holds. 
Then this should be true of all other reference- 
bodies in uniform rectilinear motion with respect 
to it. All of these would then also have to be 
regarded as Galileian, and only with such does 
the Special Theory of Relativity have to do. Or, 
the factor of gravitation, as well as those of rota¬ 
tion and acceleration, does not enter into the Spe¬ 
cial Theory. 


51 






But it is readily seen that if the theory of rela¬ 
tivity is to be universally applicable it must be 
valid for any state of motion whatever. In prac¬ 
tice, it is customary to choose axes (or planes) 
of reference on the earth perpendicular to each 
other, one north and south, one east and west, and 
one perpendicular to these. But in dealing with 
two places a considerable distance apart on the 
earth, in a north and south or east and west line, 
there is a slight rotation of the two sets of axes 
with reference to each other, as there is of the 
same set with reference to its original position 
if moved from one place to another, the rotation 
becoming complete in passing around the earth. 
And, of course, any deviation from the north and 
south or east and west line would cause a similar 
complex rotation. And the same is true in con¬ 
sidering relations between the earth and the 
heavenly bodies. Thus any conceivable rotations 
are possible, even as all conceivable relative 
velocities and accelerations are possible. 

However, in the case of any irregularity or 
non-uniformity in the motion of a body, it would 
appear that the principles so far discussed would 
not apply. The same mechanical laws would ap¬ 
parently not hold. It would therefore seem that 
non-uniform motion must have a kind of objective 
reality and could not come under the principle 
of relativity. This difficulty is, however, found 
to be only apparent; and thus for the present this 
fact may be assumed in the argument. The gen¬ 
eral principle of relativity may therefore briefly 


52 


be stated provisionally as follows: “All systems 
of co-ordinates, in any state of motion whatever, 
give us exactly the same form for the expression 
of the general laws of nature; or, a general law 
of nature ivill remain unchanged by a change in 
treatment from one co-ordinate system to an¬ 
other, in any state of motion whatever, or by a 
change of the space-time variables of one such 
co-ordinate system to those of another. 


53 




II- 

Propositions of Euclidean Geometry Not Exactly 
Valid 11 in a Gravitational Field. 

S we have seen, in a free or gravita¬ 
tionless field the results of the Special 
Theory of Relativity would hold with 
respect to a reference-body in uni¬ 
form motion in a straight line. But if 
in the same field a second reference-body, a plane 
circular disk, were rotating uniformly with respect 
to such a Galileian reference-body, an observer 
away from the center on the disk would feel a 
force radially outward. This would, of course, be 
interpreted as the effect of inertia (centrifugal 
force) by an observer at rest in free space or on 
a Galileian reference-body (as well as by one at 
the center of the disk). But by the observer 

11 Euclidean geometry as a branch of physics has its limitations, 
its truths in their application not being absolute. Propositions of 
Euclidean geometry are considered true or correct when they are properly 
derived from fundamental axioms. But this passes the question as to 
whether they are true, on to these axioms. And that these cannot be dem¬ 
onstrated by ordinary geometrical methods but are elements upon which 
Euclidean geometry is developed, is acknowledged. Hence the possibility 
of other and different systems of geometry is manifestly implied. 
While in Euclidean geometry the sum of the angles of a triangle is 
equal to two right angles, in the non-Euclidean geometries this sum is 
either more or less than two right angles. 



54 






away from the center on the disk, considering his 
disk as at rest, this force would be interpreted as 
the effect of a gravitational field acting on him¬ 
self, although the effect of this gravitational field 
would be very different from that due to New¬ 
tonian gravitation. The force would be away 
from the center and would increase from nothing 
at the center of the disk with the distance from 
the center outward toward the circumference. 
This observer could not tell, according to the 
equivalence hypothesis to be discussed later, 
whether he was at rest in such an actual gravita¬ 
tional field or whether he was located on a ro¬ 
tating body. This rotation would create the 
equivalent of a gravitational field of a particular 
kind. In like manner, it may also be said, grav¬ 
itational fields are produced by the presence of 
matter. 

If now the observer would place one clock at 
the center of the disk and another identically con¬ 
structed clock on the circumference, from the 
standpoint of a non-rotating body £ the clock at 
the center would have no velocity while the clock 
at the circumference would move relatively to S 
in consequence of the disk's rotation. Then from 
a previous consideration it would follow that the 
latter clock much go slower than the former, as 
observed from S. From this it would follow that 
the “going rate" of a clock would vary not only 
for different gravitational fields, but even for 
different positions within the same gravitational 
field. Hence gravitational fields, whether pro- 


55 


7 


duced by non-uniform motion or by the presence 
of matter, determine the rates of clocks, 12 as well 
as the states of bodies. 

So also, as judged from a Galileian system, as 
a measure applied tangentially to the edge of the 
rotating disk would be shorter than when applied 
radially, it must follow that the circumference of 
the disk thus measured divided by the diameter 
similarly measured, would not be 3.1416, but 
greater than this number. 

It is thus found from the behavior of clocks 
and measuring-rods on a rotating body that the 
propositions of Euclidean geometry are not ex¬ 
actly valid on a rotating disk, and hence not in 
any gravitational field. From the above it can 
also be inferred that the term straight line does 
not convey an absolute meaning, 13 as in a Eu¬ 
clidean sense it would not apply in such gravita¬ 
tional field, as illustrated in the case of the rotat¬ 
ing disk. Therefore the co-ordinates relative to 
such a disk cannot be defined by methods used in 
discussing the Special Theory. And before the 
co-ordinates and times of events are defined no 
exact meaning can be assigned to the natural 
laws. 

13 The above point should be borne in mind in considering: the sub¬ 
ject of space-curvature and that of the possible return of light rays 
after a cosmic curvilinear sweep through or around the space-time con¬ 
tinuum, of which we shall speak later. 

12 It is thus seen also that it is not possible from clocks arranged 
at rest with respect to a body of reference to arrive at an adequate 
definition of time. 


56 



III. 


The Necessarily Non-Euclidean Space-Time Con¬ 
tinuum and the Use of Gaussian 
Co-Ordinates . 

HE way to overcome the above diffi¬ 
culty is found in a proper under¬ 
standing of the difference between a 
Euclidean continuum and a non-Eu- 
clidean contiuum. To afford such an 
understanding an illustration of a table with a 
marble top is used. This is supposedly divided into 
an indefinite number of equal squares by rods of 
equal length. If the perfectly level marble top is of 
equal temperature throughout, any Corner of a 
little square with reference to any other corner 
may be indicated by two numbers, which then 
are the Cartesian co-ordinates of this corner. But 
if the table were heated at the middle, because of 
the expansion from the heat the construction of 
the squares would come into disorder, as the outer 
rods would not be affected by the heat. Hence, 
as a corner could no longer correctly be desig¬ 
nated by two numbers (Cartesian co-ordinates) 
with reference to another corner by the use of 
these rods, the surface must now have become a 
non-Euclidean continuum. A new system of co¬ 
ordinates, of more general application, is thus re- 



57 








quired, and this is the ingenious invention of the 
mathematician Gauss. 

As will be seen later, according to the General 
Theory of Relativity the velocity of light really 
depends on the co-ordinates, in a gravitational 
field, and can therefore not strictly be constant. 
Indeed, the presence of a gravitational field even 
invalidates the definition of the co-ordinates and 
the time at the foundation of the Special Theory. 
Hence, although the space-time continuum of the 
Special Theory may be regarded as a Euclidean 
continuum, the space-time continuum, according 
to the General Theory , cannot be Euclidean, but 
must be like that illustrated in the marble top 
with variable temperature. As Cartesian co-or¬ 
dinates cannot be applied to the heated marble 
table, so according to the General Theory a sys¬ 
tem, or reference-body, cannot be constructed 
from rigid bodies and clocks that will indicate 
position and time directly. But this difficulty is 
overcome by referring the four-dimensional 
space-time continuum to Gaussian co-ordinates, 
thus giving to every point or event of the con¬ 
tinuum four such co-ordinates. Thus the de¬ 
scription of the space-time continuum with the 
aid of a rigid reference-body is superseded by the 
description with Gaussian co-ordinate. And this 
method of description does not carry with it the 
defects of that with a rigid body of reference and 
is not limited to the Euclidean character of a con¬ 
tinuum. 

The way has now been prepared for a re- 


58 


placing of the hitherto provisional formulation of 
the general principle of relativity, which still im¬ 
plies the use of generally impossible rigid refer¬ 
ence-bodies, by an exact and workable formula¬ 
tion of that principle. The body of reference, or 
system of co-ordinates, is, in accordance with the 
above consideration, replaced by the Gaussian co¬ 
ordinate system. The general principle may now 
be stated briefly in the following words: A gen¬ 
eral law of nature will remain unchanged by a 
change in treatment from one Gaussian co-ordi¬ 
nate system to another in any state of motion 
whatever. The Special Theory of Relativity per¬ 
tains to domains without gravitational fields, a 
Galileian rigid reference-body serving as a body 
of reference, with motion relative to which the 
Galileian laws of uniform rectilinear motion of 
material points holds. By referring these gravi¬ 
tationless domains to non-Galileian reference- 
bodies a special gravitational field with respect to 
these bodies will be present. Then in such devel¬ 
oped gravitational fields there can be no rigid 
bodies with Euclidean properties, so that non- 
rigid reference-bodies, which have therefore been 
called reference-mollusks, are used, with all free¬ 
dom of motion, and thus with the possibility of 
changing in form. These are then generally 
equivalent to arbitrarily chosen Gaussian four¬ 
dimensional co-ordinate systems, all being avail¬ 
able with equal right and equal practicability in 
the formulation of the general laws of nature, but 
without affecting those laws themselves. 


59 


IV. 


An Entirely New Conception of Gravitation. 


UCH importance for the Theory of 
Relativity is attached to the fact 
that, in contrast with the action of 
electric and magnetic fields, the ac¬ 
celeration due to the sole influence of 



a gravitational field does not depend on the 
material , or on the physical state of the body 
—a fact that has been well established. The 
acceleration is found to be equal to the quo¬ 
tient of the gravitational mass divided by the 
inertial mass multiplied by the intensity of the 
gravitational field. 14 From this it is shown to fol¬ 
low that, as the acceleration does not depend on 
the material or the condition of a body and is 
always the same for a given gravitational field, the 
ratio of the gravitational to the inertial mass must 
also be the same for all bodies. 


14 Dr. Einstein’s proof may here be summarized. Let F represent 
the force, a the acceleration, i the inertial mass, g the gravitational 
mass, and f the intensity of thd gravitational field. Then, according to 
Newton’s law of motion, F = i x a, i being a characteristic constant of 
the accelerated body. And if gravitation is the cause of the acceleration, 
F — gxf, g being a characteristic constant. Therefore, ixa = gxf, or 



60 









From this it is readily deduced that the grav¬ 
itational mass is equal to (or identical with) the 
inertial mass, so that the same body manifests 
itself according to circumstances as inertia or as 
weight. By these facts not only is the general 
Theory of Relativity greatly strengthened, but the 
foundation for an entirely new view of gravitation 
is laid. If the constancy of the velocity of light 
regardless of the velocity of its source is funda¬ 
mental to the Special Theory of Relativity, then 
this equivalence of the inertial and gravitational 
mass is fundamental to the General Theory. 

In further development of the points made, we 
are asked to imagine a portion of free or gravita¬ 
tionless space, which would therefore afford the 
conditions required by the fundamental law of 
Galilei. Then if within this space there were 
placed a large room or building with an observer 
inside, for him gravitation would not exist. And 
if the room were then supposed to be caused to 
move upwards with a uniformly accelerated mo¬ 
tion as viewed from another reference-body not 
thus moved, he would feel a sense of pressure 
against the floor, or of weight. If he released any 
object whatever held in his hand it would approach 
the floor with the same accelerated relative motion. 
To this man the room and himself in it would seem 
to be in a gravitational field, although to the ob¬ 
server poised in outside space the man’s reaction 
against the floor, as well as the object’s behavior 
would be known not to be due to any so-called 
gravitational attraction, but to inertia. Being un- 


61 


aware of any motion, the man in the room might 
thus suppose himself to be motionless in a gravita¬ 
tional field. Hence what to one would be due to 
the gravitational mass would to the other be due 
to the inertial mass, and these would be the same. 

On the other hand, a man in the act of falling 
from an aeroplane or within the above room, 
falling with uniform acceleration, would not be 
conscious of any gravitation. He might also 
imagine himself to be motionless and the earth 
approaching him with his own accelerated velocity. 
Thus, if we consider the reference-body moving 
with the falling man, even as if we consider the 
reference-body stationary with the man poised in 
space in the above illustration, no gravitational 
force would be apparent. 

Moreover, by changing reference-bodies in 
both cases, that is, to the man in the rising room 
as viewed by himself, and in the case of the falling 
man to the observer on the ground, there would 
seem to be a so-called gravitational “pull.” Hence 
the sense of a gravitational field would apparently 
depend in these cases upon the point of view, or 
would be a matter of relation. This point might 
also be illustrated by the sensation of iveight in a 
rising or falling elevator. If the man were stand¬ 
ing on a platform scale and the elevator were 
rising rapidly, his weight would increase; if he 
were similarly descending, he would weigh less, 
while his weight would become zero in a freely 
falling elevator. 

Or, in general, a so-called gravitational 


62 


field could apparently be created by a mere 
change, or by a proper choice, of a body of 
reference. Thus the principle of relativity 
may be extended to include bodies of ref¬ 
erence accelerated with respect to each other. 
From the above illustrations it is seen also that 
a person on a body with any non-uniform motion 
whatsoever might even attribute the effect upon 
himself and surrounding objects to a gravitational 
field with action variable with time. 

We are now prepared for a general application 
of these principles to gravitation. Referring a 
gravitationless domain to a Gaussian co-ordinate 
system, or to a “mollusk,” as reference-body, then 
with respect to such reference-body we have a 
special gravitational field. The behavior of meas¬ 
uring-rods and clocks and freely moving material 
points with respect to this reference-body can then 
be determined by mathematical transformation 
and interpreted as due to the influence of the grav¬ 
itational field. Then from a law based upon the 
space-time behavior of this special gravitational 
field a general law of gravitation is obtained 
which also satisfies the general postulate of rela¬ 
tivity. If any matter is present in this domain, 
only its inertial mass need be considered in the 
effect in exciting a field. Now, from the General 
Theory, the influence of the gravitational field 
on the course of processes taking place according 
to known laws when a gravitational field is absent 
(Special Theory), may be determined. Thus a 
theory of gravitation is arrived at from the gen- 


63 


eral postulate of relativity that interprets the 
empirical law of the equality of inertial and grav¬ 
itational mass, which also explains certain anom¬ 
alous phenomena hitherto inexplicable. 

While Newtonian gravitation assumes a force 
of attraction between so-called mutually attract¬ 
ing bodies, Einsteinian gravitation postulates no 
such force. It appears that the presence of mat¬ 
ter causes a distortion of space, even as when a 
bullet is placed upon a stretched piece of rubber. 
Gravitation is thus supposedly due to a curvature 
or crumbling of the space-time manifold, or to 
a distortion of space and time in the presence of 
matter —to a so-called warp in space—causing 
every particle to follow a geodesic in the four¬ 
dimensional space-time manifold, instead of the 
shortest line in three-dimensional space accord¬ 
ing to Newtonian gravitation. In passing near 
a body it would pass through curved space whose 
curvature would depend upon the mass of that 
body. And thus, though its motion would con¬ 
tinue to follow its initial course, that course 
would to us appear curved or take the form of 
an orbit. If the curvature of space would be very 
small, as in the presence of but little matter, the 
geodesic would become nearly a straight line . 
which would become entirely straight in the total 
absence of matter. 

Thus if there were no matter, there would be 
no distortion of space, and hence no gravitation, 
and no aravitational field. Or, if one might still 
r.peak of space, he would have to regard it as 


64 




Euclidean; and, again, in such hypothetical 
Euclidean space there would be no gravitation. 
From this it appears that matter is not the cause 
of gravitation, but the condition, giving space the 
property called gravitation. Or, gravitation would 
be a property of space. Then to what extent can 
it be a force? 

From the illustration of the rising room it is 
seen that so-called gravitation and so-called cen¬ 
trifugal force are both proportional to the mass. 
Indeed, it was seen that the inertial mass and the 
gravitational mass are equal. Now while it is 
true that we speak of a centrifugal “force,” no 
one would seriously consider it a force. Indeed 
as is well known, so-called weight is due to the 
difference between so-called gravitational force 
and centrifugal force. Then if so-called centrifu¬ 
gal force is not a force, the natural inference 
would seem to be that that which partly balances 
it (that is, gravitation) is not a force. May not 
both, in the last analysis, then be due to inertia, 
as related to or determined by space, which would 
thus be its ultimate cause ? 

It is a remarkable fact that, in the last analy¬ 
sis, even gravitation is thus a property of space 
in the higher sense. And hence common sense 
three-dimensional space, time, inertia and gravi¬ 
tation, are only special aspects of the higher unity 
in what might be called transcendent space. 


65 


V. 


Light Deflected in Passing Through a 
Gravitational Field. 

T light is propagated curvilinearly 
passing from one medium to an- 
ler of different density, is a matter 
common knowledge; but that 
jre should be a somewhat analog¬ 
ous curvilinear propagation of light through a 
gravitational field, sounds new and strange. It 
should therefore seem quite natural that the an¬ 
nouncement of this fact by Dr. Einstein as a neces¬ 
sary inference from his general theory of relativ¬ 
ity, would excite great interest. And, indeed, noth¬ 
ing in connection with his theory has had such 
wide publicity given it as the story of the remark¬ 
able test to which this announcement was put by 
British scientists. 

This point can perhaps best be understood by 
going back to our illustration of the man falling 
from an aeroplane or within a room falling with 
uniform acceleration. To such a man there 
would be no sense of gravitation. An object re¬ 
leased from his hand would seem to him to re- 



66 






main stationary, and if he threw if with great 
force in a horizontal direction, it would from his 
point of view, if there were nothing to retard it, 
continue horizontally with uniform velocity and 
in a straight line. Nor would the nature of the 
object make any difference whatever. 

And what would be true of a material object 
would be equally true of the tip of a ray of light. 
Nor would even velocity, either in the case of the 
object or of that of the light, change this to him 
rectilinear horizontal motion. And if an object 
and a tip of light started their journey together 
in the same direction and traveled with the same 
velocity, they would continue together. To an 
observer on the earth, however, the object thrown 
horizontally by the man falling from the aero¬ 
plane or within the falling room and to whom 
there would be no sense of gravitation, as well as 
the tip of light, would follow an hyperbola. Or 
what to the man without the sense of gravitation, 
and therefore in gravitationless space, would be 
a straight line, would to the man for whom the 
gravitational field is real be a curved line. 

Thus, to make the above case general, any 
body in uniform motion in a straight line with 
respect to a Galileian reference-body, would have 
an accelerated and curvilinear motion with re¬ 
spect to an accelerated reference-body. 15 And, 
from the above consideration of acceleration and 

15 If the space-time course for any natural process, in a Galileian 
domain, relative to a Galileian reference-body, is known the appear¬ 
ance of the same as seen from a reference-body accelerated relative to 
the assumed Galileian reference-body can be determined by calculation. 


67 



gravitational action, this would, of course, also 
be true of a body passing through a so-called 
gravitational field. Thus also rays of light, which 
otherwise with reference to a Galileian reference- 
body are transmitted in a straight line, could no 
longer be strictly straight lines with reference to 
an accelerated reference-body, and therefore not 
in a gravitational field. That is, they would be 
propagated curvilinearly. Or, in other words, 
rays of light would come under the influence of 
so-called gravitation exactly the same as ponder¬ 
able matter. 16 Thus, although too small for ob¬ 
servation in ordinary cases upon the earth, for a 
ray of light passing the edge of the sun, it was 
calculated by Dr. Einstein that the deviation from 
a straight line should be 1.7 seconds of arc. Or, 
in general, at a distance of d radii from the sun's 
center, this deviation should be 1.7 seconds of arc 
divided by d. And this startling scientific predic¬ 
tion was most wonderfully confirmed by two 
astronomical expeditions, in the case of light from 
stars in the immediate field of the sun during the 
total solar eclipse of May 29, 1919, as also has re¬ 
cently again been done. 

If light were corpuscular in nature, it has been 
shown that the Newtonian law of gravitation 
would account for a deflection of .87 seconds of 
arc for a beam of light passing the limb of the 
sun. Such a possible deflection was calculated 

ia From this also it would, of course, appear' that gravitation is not 
the result of a "pull,” as it would equally affect a material body and 
imponderable light in motion. It would make gravitation truly rela¬ 
tive, or a matter of relation. 


68 



already by Cavendish in 1795. According to Dr. 
Einstein’s view of gravitation this deflection 
should be tivice that amount, or about 1.74 sec¬ 
onds of arc. Now the average deflection of rays 
passing near the sun, according to photographs 
taken during the eclipse (May 29, 1919) and 1 com¬ 
pared with photographs of the field of stars with¬ 
out the sun taken at another time, was 1.79 sec¬ 
onds of arc. This deflection for the expedi¬ 
tion to Sobral, Brazil, was 1.98 seconds, and 
that for the expedition to the Isle of Prin¬ 
cipe, West Africa, was 1.60 seconds. This aver¬ 
age is a close approximation to the results 
attained in Einstein’s calculations, and the dis¬ 
crepancy between the result obtained from 
observation and that of Einstein’s calculations, 
may easily be due either to the imperfection of 
the instruments or of their application , or to in¬ 
accuracy in Einstein’s data of calculation. At any 
rate, it has been shown that the additional deflec¬ 
tion above that due to Newtonian gravitation 
even if light were material , cannot be caused by 
refraction of the rays in passing through the 
corona of the sun. 17 

From the above it follows that the law of the 
constancy of the velocity of light in vacuo , which 

17 From this fact of deflection of light passing through a gravi¬ 
tational field it follows that the observer’s angle of displacement be¬ 
tween two stars when the sun or another star or stars comes between 
them on the observer’s side of them, would seem greater than it really 
is. Hence our calculations of distance from parallax must needs be 
more complicated, and indeed more uncertain, than has hitherto been 
supposed, especially with the possibility of unknown dark bodies exist¬ 
ing in space. Moreover, from this fact of light-deflection a single star 
might even appear double, etc. Hence the actual distance, mass, mo¬ 
tion and light of certain stars might be quite different from the re¬ 
sults reached by calculation. 



was a fundamental assumption in the Special 
Theory of Relativity, cannot be absolute accord¬ 
ing to the General Theory, as variation in direc¬ 
tion of propagation necessarily implies variation 
in velocity. Thus the velocity of light would vary 
in passing through a gravitational field, or in 
passing through parts of the same field having 
different intensity. But within a gravitationless 
field, or within a gravitational field of uniform 
intensity, it would remain constant. Indeed, be¬ 
cause of the unequal distribution of matter , and 
consequently of infinitesimal variation in the 
curvature of space , probably nothing in the uni¬ 
verse moves with strictly constant velocity, as 
will later become more apparent. 

It is thus seen that instead of the General 
Theory of Relativity overthrowing the Special 
Theory, the latter is really a special case under 
the former and holds only where the influence of 
a gravitational field would be uniform , because 
the Special Theory deals with gravitationless 
fields where it may be ignored. Moreover, as Dr. 
Einstein shows, the General Theory supplies the 
key to the investigation of the laws satisfied by 
any gravitational field theoretically obtainable. 
And from such gravitational fields of a special 
kind the general law of gravitation must be de¬ 
rivable. 


70 


VI. 


The Theory of Relativity Alone Explains the 
Anomalous Motion of the Orbit 
of Mercury. 



ALCULATIONS based upon New¬ 
ton’s laws of motion and gravitation 
are found to be only approximately 
true, although deviations from New¬ 
ton’s laws are found to be too 
small generally to be observed, owing to the 
element of but small velocity. Calculations 
based upon Einstein’s theory correct Newton¬ 
ian calculations to an astonishing degree, as 
has already been confirmed in several not¬ 
able cases. At ordinary velocities, however, 
the results of the theory of relativity become vir¬ 
tually identical with those of the mechanics of 
Galilei-Newton. As applied to the perplexing 
case of Mercury’s orbit, Dr. Einstein has shown 
that the General Theory of Relativity almost ex¬ 
actly accounts for its anomalous rotation after 
corrections for known causes have been made. 

The perihelion of the orbit of Mercury has 
for many years been known slowly to advance in 


71 




the direction of the planet’s revolution, the ob¬ 
served amount being 574 seconds of arc per cen¬ 
tury. Of this amount 532 seconds have been cal¬ 
culated to be due to the influence of the other 
planets. Thus there has remained an advance of 
42 seconds to be accounted for. For a long time 
this was believed to be due to some undiscovered 
planet within the orbit of Mercury; but all search 
for the invisible disturber has failed. Now, ac¬ 
cording to the theory of relativity, Dr. Einstein 
has calculated that the perihelion point of that 
orbit should advance just 43 seconds of arc more 
than can be accounted for by Newton’s law of 
gravitation. The difference between this calcu¬ 
lated amount of 43 seconds and the observed 
amount of 42 seconds may easily be due either to 
incorrect data in the calculation or to inaccuracy 
in observation. It might be stated that as the plan¬ 
et’s velocity is greater at perihelion, because of 
its closer proximity to the sun, its energy or mass, 
as has been shown, would therefore be greater, 
and hence its gravitational field would be greater, 
however minutely so. Thus its velocity would 
also be increased. 

Moreover, the ellipses of all planets must ro¬ 
tate in the same manner, although because of 
their closer approximation to a circle and their 
slower motions due to greater distances from the 
sun, this rotational advance would be impercepti¬ 
bly small. And thus this explanation of a phe¬ 
nomenon of the rotation of Mercury’s orbit, 
which had baffled astronomers since the time of 


72 


Leverrier, affords another astonishing confirma¬ 
tion of Einstein's General Theory of Relativity. 

In order not to make a separate division of it, 
we shall here also mention a third remarkable 
inference from the General Theory of Relativity. 
If we regard an atom emiting spectral lines as 
a clock, its environing gravitational field would 
determine the frequency with which it emits or 
absorbs light. Thus on the surface of a large 
body like a star the frequency of an atom would 
be somewhat different from that of an atom of 
the same element in a free state or on a small 
body. For the sun the displacement toward the 
red end, it has been calculated, should be about 
two-millionths of a wave-length of light. Hence 
spectral lines of light reaching us from largo 
masses like distant stars should be displaced to¬ 
ward the spectrum’s red end of slower vibration 
as compared with the corresponding lines for 
light produced terrestrially in a similar manner. 
And even this is reported to have been verified 
by several investigators, while some have, how¬ 
ever, not found this effect. Thus, as this displace¬ 
ment of spectral lines must vary with the magni¬ 
tude of bodies, it may some day furnish the 
astronomer with important data for the calcula¬ 
tion of the mass of stars. 


73 


VII. 


The Physical Universe a Finite Entity. 

T has been contended for some time 



by a number of writers, as the most 
natural conception of the universe, 
that it is infinite in extent of space 
with a somewhat variable distribu¬ 


tion of matter throughout. But for such a universe 
it has been shown, gravitation would approach 
and ultimately reach infinity with the indefinite in¬ 
creasing of the radius from a chosen center. 
However, such supposedly infinite gravitation 
would not only be impossible, but it would in¬ 
volve some insurmountable difficulties. 

The Newtonian theory of gravitation, as has 
been pointed out, demands some kind of center, 
from which in every direction outward the density 
of matter (distribution of stars) should dimin¬ 
ish until it would end in' an infinite region of 
emptiness. But according to such a conception, 
light, and perhaps certain stars, would perpetu¬ 
ally pass outward into supposedly unoccupied 
space never to return. On the other hand, accord¬ 
ing to the General Theory of Relativity with its 
non-Euclidean space, which has already been dis¬ 
cussed, or what might almost be called spherical 
space, straight lines would at first diverge, and 


74 




then converge to a “counter-point” (or counter¬ 
points) of the starting point. Thus, for all we 
know, some objects that appear to us as stars 
might be counter-stars (images) of stars on the 
opposite focal-point of space. This might be con¬ 
sidered as a suggestive explanation of so-called 
negative parallax, thus indicating a star in the 
diametrically opposite direction. In such curved 
space we might, if we could look far and deep 
enough, look back upon ourselves through or from 
the opposite point of the universe. Then how 
about time, if it is inseparable from space? How 
about past and future? 

On a spherical surface two lines apparently 
straight to a two-dimensional being, but really 
following great circles on a sphere to three-dimen¬ 
sional beings, if drawn from a common point, 
diverge until they reach their maximum diverg¬ 
ence from each other at points one-fourth of a 
great circle from their starting point. After¬ 
wards they converge until they meet on the oppo¬ 
site side of the sphere. And if they were contin¬ 
ued from that meeting point, each would return 
on or coincide with what might be considered its 
prolongation through the original starting point. 
Even two-dimensional beings might discover cer¬ 
tain of these facts. As we have noted might have 
been done by early human investigators, they 
might even by actual measurement discover that 
while the ratio of the circumference to the diam¬ 
eter for small circles drawn on the earth is 3.1416, 
for sufficiently large circles thus drawn, that ratio 


75 


would not exactly hold, and that there is some 
definite law of decrease in that ratio with the 
increase of the circle. 

Hence such beings might mathematically ar¬ 
rive at the conclusion that their apparently flat 
two-dimensional universe might curve into a 
third dimension of what to them would be trans¬ 
cendent space. If they had made enough prog¬ 
ress in discovery to ascertain certain to us well- 
known facts, and if their mathematics were suf¬ 
ficiently developed to make the necessary calcu¬ 
lation, they might even determine, though they 
could not conceive, the shape and size (radius) of 
their “universe,” as a to them transcendent 
sphere. But if such beings were confined to a rel¬ 
atively very minute area of the spherical surface, 
it might be difficult, if not impossible, for them 
ever to rise above their belief that their “uni¬ 
verse” must be an infinitely extended plane sur¬ 
face. 

What would be true of such two-dimensional 
beings within a limited area on a spherical world, 
is apparently true of us three-dimensional beings, 
confined within a little corner of the great whole 
of the universe. A curved space is as inconceiv¬ 
able by us as a curved surface would be by two- 
dimensional beings. And yet the theory of rela¬ 
tivity gives us data analogous to those within the 
reach of the two-dimensional beings spoken of 
above, which indicate a curvature of space and 
what might almost be considered a sort of spheri¬ 
cal and finite universe, and which even afford some 


76 


basis for a theoretical calculation of its possible 
shape and size . 

An entity can be finite and yet unbounded. A 
circle or curved line with ends connected, though 
limited in the area it encloses, is nevertheless 
endless as a line. While a line in itself has only 
one dimension, its curvature, which in the above 
case makes it seemingly endless, is in the second 
dimension. So a surface which itself has only 
two dimensions, by a curvature in the third di¬ 
mension so as to form a sphere or other similar 
body, becomes apparently endless even though its 
area remains finite. And a being whose habitat 
that sphere might be, would necessarily always 
be confined to it, however much it would wander 
about even so as to return to its starting point. 

And so our universe of so-called three-dimen¬ 
sional space in curving into a fourth dimension 
would likewise have to be finite and yet also be 
boundless. So-called straight lines in such a uni¬ 
verse would return in upon themselves, even as 
would be the case on the surface of a sphere. In¬ 
deed, it is curvature in an added dimension that 
causes otherwise apparently possible infinites to 
be finite, whether in the case of a line, a surface, 
or a so-called solid or even the universe itself. We 
might even add that, as there is probably no mo¬ 
tion strictly in a straight line and uniform in the 
universe, as we have already pointed out, due to 
the presence and unequal distribution of matter 
throughout it, it could not be infinite in extent ac¬ 
cording to the principle just set forth. 


77 



Indeed, as already noted, according to the 
General Theory of Relativity, space is not an in¬ 
dependent entity, but is determined by matter. 
All speculations concerning space must therefore 
be based upon the state and amount of so-called 
matter (or energy) , 18 The very existence of space 
and time is associated with or dependent upon the 
existence of matter. In the scientific structure of 
Galilei-Newton, space and time had an absolute 
existence and were distinct from each other. They 
were regarded as though they were the contain¬ 
ers for matter with its motions and other activi¬ 
ties within the universe, space to afford the pos¬ 
sibility of separation and extension and time to 
afford duration and interval. Hence time and 
space supposedly had an existence even apart 
from the physical universe, and the very exist¬ 
ence and activity or persistence of matter were 
supposedly dependent upon them. 

In the scientific structure of Dr. Einstein, space 
and time are wholly relative, and are depend¬ 
ent for their existence upon matter with its activ¬ 
ities. Instead of being self-existent containers 
of an infinite extent, .their existence and extent 
are determined by matter , without which they 
would cease to be. Matter thus makes space-rela¬ 
tions possible, and the density or relative amount 
of matter determines the magnitude of the latter. 
Indeed, we acquire our conception of space en- 

18 Speaking in terms of volume, the extent of matter in our galactic 
system may be to its spatial extent approximately as 1 to 174 x 10 21 . 
See the author’s “Whence Came the Universe,” pp. 158-160; also pp. 
162-165, where four-dimensional space is considered in proof that the 
physical universe must be finite. 


78 



tirely from matter, or matter relatively in motion. 
And events in a material domain give meaning 
and measure to time. Hence as we have seen all 
motion to be relative, and indeed wholly imagi¬ 
nary apart from matter and some body of ref¬ 
erence, so must all space be relative, and without 
matter it must be wholly imaginary or simply 
an abstraction. 

As the velocities of stars are small as com¬ 
pared with that of light, the nature of the uni¬ 
verse as a whole with matter relatively at rest 
may, as already intimated, from this also be deter¬ 
mined. 

We have seen that Euclidean geometry is not 
exactly valid in its application to the universe, 
because of the presence and distribution of mat¬ 
ter. But although the universe deviates but 
slightly from the Euclidean, it cannot be a quasi- 
Euclidean entity, one only slightly in parts devi¬ 
ating from the Euclidean like a billowy surface. 
This appears from the fact that the space of such 
should be infinite —which is impossible according 
to the above consideration—with zero as a result¬ 
ant density of matter, which is contrary to ob¬ 
served fact. Hence a universe to have any density 
of matter greater than zero , as observation shows 
it has, would have to be what might for want of 
a proper term be called spherical but with un¬ 
equal deviations due to unequal distribution of 
matter. 

As matter is not arranged in space exactly 
according to the laws of Euclidean geometry of 


79 


solids, there must be a so-called “warp in space” 
and from any one point of view an indefinite num¬ 
ber of partial warps. Hence its general devia¬ 
tion or warp from what might be called Euclidean 
rectilinear space would indicate the so-called gen¬ 
eral form and become the approximate measure 
of the extent of the universe, while the particular 
or partial warps would indicate its deviations 
from the general form. And, of course, all would 
be due to the distribution of matter with its so- 
called gravitational fields and what might be 
called its general gravitational totality. And such 
a universe must necessarily be finite. The very 
presence of matter, however rarified or attenu¬ 
ated and however unevenly distributed, would 
make the universe a finite entity. 

In the universe with its distribution of matter 
as we know it, it can be calculated how much 
the space-time path should deviate from a so- 
called straight line. And however large or small 
the universe might thus be, nothing, not even 
light, could pass beyond it. It would continue 
its course within it in what would seem to us to 
be a circle, back ^to its origin, etc. 18 * Hence, as 
has been suggested, we might see the light of cos¬ 
mic events that took place a universe-light-cycle 
ago. Nor could there be anything physical be¬ 
yond such a universe, unless it were another sim- 

ls * From one calculated amount of the matter in the universe as we 
know it and the relative amount of space-curvature to mass according to 
this wonderful theory, the extent of the universe should approximately 
require light 625,000,000 years to return to its starting point. Or, 
speaking in terms of Euclidean measurements, it might be said to be 
about 200,000,000 light years in diameter. 


80 



ilar universe or number of universes so created 
as to be wholly independent of our universe or 
of one another. And even then in their aggregate 
they would still be finite, and this would of course 
be true of their matter and consequently of their 
so-called space (“Whence Came the Universe?” 
pp. 146-152). 

The above is thus a striking confirmation of 
conclusions reached by another process of rea¬ 
soning, namely, that a physically constituted uni¬ 
verse cannot be infinite, and must therefore be 
relative and dependent as a whole and interde¬ 
pendent in every part from electron to star. 1 * 
Hence the Einstein theory of relativity within the 
existing universe also unmistakably points to a 
dependence of that physically finite universe to 
an infinite Entity wholly different in essence (an 
independent or absolute spiritual Personality) 
superior to it, both immanent and transcendent. 
The theory of physical relativity in the parts thus 
necessarily implies, and indeed is of a piece with, 
that of spiritual dependence of the ivhole. Hence 
a finite and temporal, and therefore created, uni¬ 
verse issues also from this latest scientifico-phi- 
losophic world-view. 

19 Creation Ex Nihilo: The Physical Universe a Finite and Tem¬ 
poral Entity. By L. Franklin Gruber. Boston. Richard G. Badger. 
1918. Reissued, 1921, under the title. Whence Came the Universe? 


81 



INDEX 


Age, relativity of, 36. 

Apologetic value of Theory 
of Relativity, 11, 12, 81. 
Cartesian co-ordinates, 20, 21, 
57. 

Cavendish, on deflection of 
light, if corpuscular, 69. 

Clocks, apparently go slow¬ 
er when in motion, 32, 35. 

Contraction, theory of Lo- 
rentz and Fitzgerald, 25, 33; 
explanation by Einstein, 33; 
amount of, 32-34. 

Co-ordinates, Cartesian, 20, 
21, 57; four necessary to mark 
event in space-time, 29; Gaus¬ 
sian, 57-59. 

Copernican system, 7, 16. 

Curvature, adds a dimen¬ 
sion, 46; of space, 64; due to 
presence of matter, 64; in¬ 
volves finiteness, 77. 

D’Alembert, on time as 
fourth dimension, 42. 

Distance, relativity of, 28. 

Earth, diurnal and annual 
motions of, 17; impossibility 
of directly determining mo¬ 
tions of, 17; physical laws un¬ 
affected by uniform motions 
of, 18; motion of, with sun 
through space, 21. 

Einstein, Dr. Albert, 8, 9, 
24ff., etc. 

Energy, relativity of, 39; not 
a constant quantity, 39; law 
of conservation of, only rela¬ 
tively true, 40; identical with 
mass, 40. 

Ether, relation of, to the 
earth, 23; does not drift with 
the earth, 23; Michelson and 


Morley, Morley and Miller, 
experiment as to hypothetical 
drift of, through the earth, 
24; not needed in Einstein 
Theory, 25. 

Euclidean geometry, not 
valid in gravitational field, 

54, not in a gravitational uni¬ 
verse, 79; based upon undem¬ 
onstrated axioms, 54; non¬ 
universality of application of, 
illustrated with rotating disk, 

55, 56. 

Event, description of an, 

29. 

Forerunners of Dr. Ein¬ 
stein, 9. 

Foucault’s experiment, 18. 

Four - dimensional space, 
mathematically represented, 
29 ; 30, 42; conception of, dif¬ 
ficult, 41, 42; free will and, 
43; super-mind and, 43; and 
three-dimensional being, 42- 
46; and four-dimensional be¬ 
ing, 46; explained and illus¬ 
trated, 43-45. 

Fourth dimension, 34, 36, 
41; time as, 29, 30, 41-43; and 
three-dimensional beings, 42- 
46. ' 

Galileian “transformation,’’ 

30. 

Galileo, recantation of, 16. 

Gaussian co-ordinates, 57- 
59. 

General Theory of Relativ¬ 
ity, stated, 53, 59; space-time 
of, non-Euclidean, 58. 

Geodesic, followed by every 
particle in the space-time 
manifold, 64. 


82 


Gravitation, velocity of ac¬ 
tion of, not greater than that 
of light, 40; new conception 
of, 60; theory of, developed, 
63; due to curvature of space, 
64; difference between New¬ 
tonian theory and Einstein- 
ian theory of, 64; a property 
of space, 65; not a force, 65. 

Gravitational field, artifi¬ 
cially formed on rotating 
disk, 66, in elevator, 61, 62, 
ingeneral by change of body 
of reference, 63; influence of, 
on rates of clocks and states 
of bodies, 55, 56; development 
of, 59; causes displacement of 
spectral lines toward red end, 
73. 

Gruber, L. F., on ratio of 
volume of matter to that ot 
space in known universe, 80; 
on finiteness of universe and 
its dependence upon an in¬ 
finite Personality, 81. 

Higgins Prize Contest, 8. 

Infinity of universe, difficul¬ 
ties in conception of, 74; New¬ 
tonian gravitation involving 
contradictions as to, 74. 

Instantaneous action, not 
possible, 40. 

Light, velocity of, 24; ve¬ 
locity of, the limiting velocity, 
32, 35, 36, 40; propagated 
curvilinearly through non- 
uniform gravitational field, 
66; explanation of deflection 
of, 67, 68; amount of deflec¬ 
tion of, 68; verification of cal¬ 
culated amount of deflection 
of, 68, 69; deflection of as a 
factor in determining mass 
and distance of stars, 69. 

Limiting velocity, velocity 
of light the, 32, 35, 36, 40. 


Lorentz and Fitzgerald, con¬ 
traction theory of, 24, 25. 

Lorentz “transformation" 
30; as proof of inseparable¬ 
ness of time and space, 30, 31. 

Mass, relativity of, 39; not 
a constant quantity, 39; law 
of conservation of, only rela¬ 
tively true, 40, identical with 
law of conservation of energy, 
40; identical with energy, 40; 
gravitational and inertial, 
equal or identical, 60, 61. 

Matter, as determining 
space and time, 78 . 

Measuring-rods, apparently 
shortened when in motion, 32. 

Meeting of Royal Society 
and Royal Astronomical So¬ 
ciety of Great Britain, 8. 

Mercury, anomalous motion 
of, accounted for by Einstein 
Theory, 71, 72. 

Michelson and Morley, Mor_ 
ley and Miller, experiment, 
24; explanation of results of, 
by Lorentz and Fitzgerald, 24. 

Minkowski, on “world-line," 
29; on four-dimensional 
world, 41, 42. 

Motion, uniform, impossible 
directly to detect by person 
on moving body, 18; physical 
laws unaffected by. 18-21; rel¬ 
ativity of, 19-22; old view of 
absoluteness of 19; of sun 
and stars, 21; everything in, 
22; no fixed point from which 
to determine, 22; non-uni¬ 
form, and physical laws, 52; 
Mercury's anomalous, ac¬ 
counted for, 71, 72; no, in a 
straight line nor uniform, 78. 

Negative parallax, possible 
explanation of, 75. 

Personality, as necessary 


83 


Creator of universe, 12. 81; 
as highest creature, 12, 43. 

Plunck, Dr. Max, on Eein- 
stein Theory, 7. 

Reciprocal relation of space 
and time, 36, 37, 41. 

Relativity, of motion, 19- 
22, 27; of simultaneity, 26; of 
time, 26, 78; of distance or 
length, 27; of space, 28, 78; 
of mass, 28; of energy, 28, 39; 
of length of measuring-rod, 
32; of going-rate of clocks, 
32, 35; of age of person, 36; 
of universe as a whole to the 
Great Absolute, 81. 

Shortening of bodies, ap¬ 
parent, 32-34. 

Simultaneity, relativity of, 
26. 

Space, relativity of, 20, 78, 
79; no absolute, 28; varying 
inversely with time, 36, 37; 
inseparable from time, 41; 
curvature of, 64; determined 
by matter, 78, 79. 

Space-time, space and time 
different aspects of, 28; space 
and time local associations of, 
28, not separable in full des¬ 
cription, 28; conservation in, 
37; “interval” in, absolute or 
constant, 37; differently di¬ 
visible into space and time, 
37; a four-dimensional con¬ 
tinuum, 41, as such non Eu¬ 
clidean, 57; gravitation a 
property of, 65. 

Special Theory of Relativi¬ 
ty, statement of, 30; space- 
time continuum of, Euclidean, 
58; a special case under the 
General Theory, 70. 

Spectral lines, displaced 
toward red end with intensity 
of gravitational field, 73; dis¬ 


placement of, a factor in cal¬ 
culating of mass of stars, 73. 

Thompson, Sir J. J., on the 
Einstein Theory, 7. 

Three-dimensional and two- 
dimensional beings, 44, 75, 76. 
Three-dimensional beings, de¬ 
termining of four-dimensional 
world by, 76, 77. 

Time, relativity of, 26, 78, 
79; no absolute, 28; as a co¬ 
ordinate, 29, 30; as fourth di¬ 
mension, 29, 30, 41-43; in¬ 
separable from space, 41; 
why not conceivable as dimen¬ 
sion, 42; has no existence 
apart from matter, 79. 

“Transformations,” ’ Gali- 
leian and Lorentz, 30. 

Two-dimensional beings, 
determining of three-dimen¬ 
sional world by, 75, 76. 

Universe, early crude con¬ 
ceptions of, 15, 16; every one 
the center of own, 37; a finite 
entity, 74, 78-80; contradic¬ 
tion in Newtonian conception 
of as infinite, 74; consistency 
of Einsteinian conception of 
as finite, 74, 75; relative ve¬ 
locity of matter in, 78; non- 
Euclidean, 79; approximate 
extent of, 80; necessarily 
caused by a spiritual Person¬ 
ality. 

Velocity of light, 24; con¬ 
stancy of, in gravitationless 
and uniformly gravitational 
fields, 24, 25; not constant in 
non-uniform gravitational 
field, 66. 

“World-line” of Minkow¬ 
ski, 29. 

World-view, new, strange, 
17, 46; difference between 
Einsteinian and Newtonio- 
Euclidean, illustrated, 37, 38. 
















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